models6'7 have illustrated the different behaviour of h.v.d.c. links and the complex nature of a.c./d.c. load flow solutions. The active and reactive power transfers ...
Integration of h.v.d.c. links with fast-decoupled loadflow solutions Prof. J. Arrillaga, M.Sc. Tech., Ph.D., C.Eng., F.I.E.E., and P. Bodger, B.E.
Indexing terms: D. C. power transmission, Numerical methods, Power system analysis computing Abstract A model of the high-voltage direct-current transmission link suitable for incorporation in fast decoupled a.c. loadflow programs is described. The model is not restricted to a particular control mode and provision is made to alter the control equations according to prespecified constraints for the variables. The versatility of decoupled programs is increased with this addition and it is shown that their reliability and computational efficiency are maintained.
VLO
v Ep d L(
I Y B G a X a
5 Rdc
R X
P Q
AP
Ag VL\1I
Xm
Xn
1
= = = = = = = = = = = = = = = = = = = =
nodal voltage (phase angle referred to slack node) direct voltage convertor terminal a.c. voltage alternating current (r.m.s.) direct current admittance susceptance conductance transformer ratio reactance delay angle extinction angle d.c. line resistance residual d.c. link variable active power reactive power active power mismatch reactive power mismatch convertor node voltage (phase angle referred to convertor current) = commutation reactance at the rectifier end = commutation reactance at the invertor end Introduction
Modifications of the generalised Newton-Raphson solution based on the decoupled principle1 are gradually displacing numerical techniques previously used for the solution of the nonlinear equations involved in power flow calculations. The efficiency of the basic fastdecoupled algorithm2 is unlikely to be substantially improved but some useful additions have been recently suggested3 for the representation of particular items of plant. An item of plant without suitable representation is the highvoltage direct-current transmission link. The omission is often justified by the scarcity of such links and even more often by the difficulties involved in their modelling. Earlier4's and recent models6'7 have illustrated the different behaviour of h.v.d.c. links and the complex nature of a.c./d.c. load flow solutions. The active and reactive power transfers to and from nodes connected to h.v.d.c. links do not obey the general rules of a.c. power transmission, i.e. the active power is independent of phase-angle relationships and the reactive power, although affected by, is not directly related to, voltage variations-.- Under such circumstances it is difficult to visualise h.v.d.c. models compatible with the behaviour of the fast-decoupled algorithm. However, acknowledging the general acceptance of fast-decoupled programs and -the existence of some h.v.d.c. links (particularly considering their large power carrying capacities), a model of the h.v.d.c. transmission link suitable for incorporation within fastdecoupled load flow programs is described in the following sections. 2
D.C. link equation
With reference to Fig. 1 and Appendix 10.1, the following relationship can be written between the voltages on the a.c. and d.c. sides of convertor (V): Paper 7853 P, first received 13th July 1976 and in revised form 6th December
1976 Prof. Arrillaga is with the Department of Electrical Engineering, University of Canterbury, Christchurch 1, New Zealand, and Mr. Bodger is with the New Zealand Electricity Department
PROC. IEE, Vol. 124, No. 5, MAY 1977
0)
Vdm = KlEm COS0m
List of principal symbols
Similarly, for the currents (in per unit)
Taking the a.c. current as a phase reference and ignoring the resistance of the transformer, the following relationships can be written for the real and imaginary components at the rectifier end (m): Kit* = BmEm sin 0 m - B m a m Vm sin \JJT
(2)
0 = Em cos m ~amVm cos i//m or in terms of the d.c. voltage: 0 =
Vdm-KlamVmco%iim
(3)
Similar equations apply at the inverter end (with opposite sign in eqn. 2). Finally, the following three equations can be written relating the direct voltages and currents: Vdm =
cosa m
-K2XmId
(4)
Vdn=
KxEncosbn-KzXnId
(5)
V
= R I
(6)
— V
For optimised d.c. power flow conditions, the control angles am and 8n will be the minimum (specified) values and the convertor voltage control will be achieved by transformer tap variations. Eqns. 1—6 contain 13 variables as follows: Vdm, Vdn,Em,En,m,(t>n,otm, 8n,am,an,
i//m, i// n ,/ d .
To eliminate trigonometrical nonlinearity and avoid overflows with infeasible operation modes, cos am and cos 5 m are used as variables instead of am and 5 n . To solve for the 13 variables, four equations or control specifications are required. These are normally the direct current ld (or the d.c. power), the optimum values of the control angles (cos m
~AnBn Encu>n ~AmBm
~AnBn cu>n [BB"]T =
EmBm AnEnBn
A
F
,Bm
~AnBn
EnBn
+ 2A»nBm AnEnBn
vm 1A nBn
vn
—KIJ 1
AmBm
COS \j/Tn
sin \pm
~AnBn sint//n
cos i//n
dependent on control equations where
xcom = sin (i// m — 0 m ) , scj n = sin (i// n — 0 n ) cco m = cos(i// m - 0 m ) , c a > n = cos(i// n - 0 n )
PROC. IEE, Vol. 124, No. 5, MAY 1977
467
10.5
/7-matrix equations
D.C. link R m a t r i x with convertor control angles, d.c. link ,. , .-. , °^ voltages and power specified. _, ,. w , T, Ri = Vdm - A j ^ cos0 m *
